Author(s) Mukoyama, Takeshi; Sarkadi, Citation University (1987), 64(5-6):

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Chemi University (1987), 64(5-6): 307-311 Issue Date 1987-03-03 URL http://hdl.

1 Title Approximate Continuum Wave Function Ionization in Ion-Atom Collisions Author(s) Mukoyama, Takeshi; Sarkadi, Lászlo Citation Bulletin of the Institute for Chemi University (1987), 64(5-6): Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 Bull. Inst. Chem. Res., Kyoto Univ., Vol. 64, No. 5-6, 1986 Approximate Continuum Wave Function for Inner-Shell Ionization in Ion-Atom Collisions Takeshi MUKOYAMA* and Laszlo SARKADI** Received September 25, 1986 An approximate electronic wave function for final state in inner-shell ionization during ion-atom collisions has been tested. The continuum wave function with zero kinetic energy is expressed in terms of the Bessel function and compared with the exact continuum wave functions with various kinetic energies and orbital angular momenta. The applications of this wave function to the calculations of inner-shell ionization cros sections are discussed. KEY WORDS: Ion-atom collision/ Inner-shell ionization/ Continuum wave function/ I. INTRODUCTION The inner-shell ionization is one of the most interesting processes in ion-atom collisions. Various theoretical models have been developed to calculate the innershell ionization cross sections by heavy charged-particle impact. It is well known that the plane-wave Born approximation (PWBA)1) and the semiclassical approximmation (SCA)2) give the theoretical cross sections in good agreement with the experimental values. In most theories, the hydrogenic wave functions have been employed. However, owing to the final continuum state, the calculation of ionization cross sections is in general more tedious and time-consuming than that of excitation cross sections. The electronic wave function for the continuum state in the nuclear Coulomb field is usually expressed in terms of the confluent hypergeometric function, which is not so convenient to evaluate the transition matrix element. We have introduced the approximate continuum wave function in our previous work for the second-order SCA calculations of L-shell ionization cross sections by heavy-ion bombardments.3) This wave function corresponds to the continuum Coulomb wave function with zero kinetic energy and is expressed in terms of the Bessel function. It is the purpose of the present paper to test the validity of application of this approximate wave function to the calculation of inner-shell ionization cross sections. For this purpose, the approximate wave function is compared with the exact Coulomb continuum wave functions with various kinetic energy and orbital angular momenta. * rh] 1J : Laboratory of Nuclear Radiation, Institute for Chemical Research, Kyoto University, Kyoto 606. ** Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI), Debrecen, Hungary._ ( 307 )

3 T. MUKOYAMA and L. SARKADI II. CONTINUUM COULOMB WAVE FUNCTION The continuum wave function with momentum k in the Coulomb field of the nucleus with nuclear charge Z is given by4) 01(r) =k rt F1( Z/k, kr),(1) where Nk1 is the normalization constant, r is the radial distance of the electron from the nucleus, 1 is the orbital angular momentum, and F1(27, p) is the Coulomb wave function.5) Atomic units (m=ii=e=1) are used. The Coulomb wave function can be written in terms of the confluent hypergeometirc function and computed by means of a series-exapnsion method.5) On the other hand, the continuum wave function with zero kinetic energy in the Coluomb field is expressed by3,6) 01(r) = Z3/2 (2/0)112 J21+1{(8p)112],(2) where p =Zr and J, (x) is the Bessel function of order 1. III. RESULTS AND DISCUSSION The comparison of the approximation wave functions with the exact ones is t. _E ~'L EXACT1 EXACT ab -1-1 ill,..0`0/ 0.01E /LO r/z (a. u. )r/z lo. u. ) O M ^-/\\/ _ d EXACT / C_ RPPRO/ 1 0 \/ `_. - - RPPRO/n.l_\ \0 /M--\/ L -/ E 0.10\/ E / L 0 \ / L 0 /- -1IV1I _ l I I I- -I r/z (a.u. )r/z (o.u. ) Fig. 1. Comparison of approximate continuum wave function with exact wave functions for 1=0. The solid line represents the exact wave function and the dashed line indicates the approximate one. a) E=0.01 Ry, b) E=0.05, c) E=0.1, and d) E=0.5. (308)

4 Approximate Continuum Wave Function for Inner-Shell Ionization made graphically for 1=0, 1, and 2. The exact wave functions with kinetic energies E=0.01, 0.05, 0.1, and 0.2 Ry are considered, because in the inner-shell ionization the ejected electrons with small kinetic energies are important. As can be seen from Eq. (2), the approximate wave function, as a function of r, depends on Z and 1. It is well known that the first-order PWBA cross section and the SCA cross section with straight-line trajectory show a universal behavior according to Z. Considering this fact, the wave functions are calculated as a function of r/z, instead of r. The calculated results of rcbl(r/z)/z for 1=0, 1, and 2 as a function of r/z are shown in Figs. 1, 2, and 3, respectively. As expected from the assumption for the approximate wave function, it can be seen that for low enregies, ES0.05 Ry, the approximate wave function is in good agreement with the exact wave functions. When the energy becomes higher, the exact wave function oscillates more quickly than the approximate one. It is also clear from the figures that at small radial distances, r/z<4.0, a.u., the approximate wave function reproduces well the behavior of the exact wave function. If the dominant part of the transition matrix element for the ionization process comes from this region, the use of the present approximate wave function for the calculation of ionization cross sections is quite adeaquate. This corresponds to the case of EXACT'-' _\- EXACT j_- - RPPROj_\- - APPRO op ao0b \ \ 1M-M-\ -] -/-1 _/ LL E E 0.05 L _I.. L 19L _ ri I _2 I I I I tl r/z (a.u. )r/z (a.u. ) \ ^ _\ EXACT_\ \ - - RPPROj _\ _ EXACT O 0 -C \\0-d \ - RPPRO \\ -1 /..\ L_/ - E 0 L-.10E 0.20 \ 3 -L 13L 1\ -2 r 1 1 I 1 _1. I, I I, i r/z (a.u. )r/z (a.u. ) Fig. 2. Same as Fig. 1, but for 1=1. (309)

5 T. MUKOYAMA and L. SARKADI \_ \b _ \01 -/N \ \ NJ _ / \ _- EXACT \^ -- - / - - EXACT (*PRO \ \ -- - PPPRO\M0\ Ḷ 9E 0\L 0.01\ ;3 L - L 2L _ - L E I I C I 1 I L r/z (a. u. )r/z (a. u. ) ~i~~a1 O1 -/- -/~\0/\\ EXACT /\_- - - PPPRON M-/-EXACT\N.1 0. RI -/ - - PPPRO \ ~.1-0\ LL 3 - E 0.10;-1 - E 0.20 L _L2L _ L 2-1litItitij -2 I I, 1 1 I Ll L! r/z (a. u. )r/z (a. u. ) Fig. 3. Same as Fig. 1, but for 1=2. ionization by low-energy projectiles. For larger 1 values, the discrepancy between the approximate and exact wave functions arises at smaller r. For fixed values of r, the discrepancy is larger for larger 1 than for small 1. Since high partial waves are important when the projectile energy is high, their contributions to the inner-shell ionization cross sections for low-energy projectiles are usually small. This fact also indicates the validity of the present approximate wave function to the inner-shell ionization process by low-energy projectiles. In conclusion, we have compared the approximate Coulomb wave function with zero kientic energy with the exact Coulomb continuum wave functions in the lowenergy region. At low energies, E<0.05, the present wave function is a good approximation to the exact continuum wave function. It is found also that the approximate wave function can reproduce well the behavior of the exact one for small radial distances, r/z<4.0. The agreement is better for smaller 1 values. These results indicate that the present approximation is useful to calculate inner-shell ionization cross sections by low-energy projectiles. The advantage of the present approximate wave function consists in the fact that the matrix element for the ionization process can be calculated analytically in the form of simple algebric functions, as has been shown in our previous work.3) It (310)

6 Approximate Continuum Wave Function for Inner-Shell Ionization are made with the present approxi- is hoped that more PWBA and SCA calculations mate wave function for the final state. REFERENCES (1) E. Merzbacher and H. W. Lewis, Handb. Phys., 34, 166 (1958). (2) J. Bang and J. H. Hansteen, K. Danske Vidensk. Selsk. Mat. Fys. Medd., 31, No. 13 (1959). (3) L. Sarkadi and T. Mukoyama, Nod. Instr. and Meth., B4, 296 (1984). (4) H. A. Bethe and E. E. Salpeter, "Quantum Mechanics of One- and Two-Electron Atoms," Plenum/Rosetta, New York, (1977). (5) M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions," Dover, New York, (1970). (6) L. D. Landau and M. E. Lifshitz, "Quantum Mechanics," 2nd ed., Pergamon, New York, (1965), Ch. 5. (311)

Title in the Plane-Wave Born Approximatio. Author(s) Mukoyama, Takeshi; Sarkadi,

Title in the Plane-Wave Born Approximatio. Author(s) Mukoyama, Takeshi; Sarkadi, Title A Computer Code for K- and L-Shell in the Plane-Wave Born Approximatio Author(s) Mukoyama, Takeshi; Sarkadi, László Citation Bulletin of the Institute for Chemi University (1980), 58(1): 60-66 Issue